We investigate topological signatures in the short-time nonequilibrium dynamics of symmetry protected topological (SPT) systems starting from initial states which break a protecting symmetry. Naively one might expect that topology loses meaning when a protecting symmetry is broken. Defying this intuition, we illustrate, in an interacting Su-Schrieffer-Heeger (SSH) model, how this combination of symmetry breaking and quench dynamics can give rise to both single-particle and many-body signatures of topology. From the dynamics of the symmetry broken state, we find that we are able to dynamically probe the equilibrium topological phase diagram of a symmetry respecting projection of the post-quench Hamiltonian. In the ensemble dynamics we demonstrate how spontaneous symmetry breaking (SSB) of a protecting symmetry can result in a quantized many-body topological “invariant” which is not pinned under unitary time evolution. We dub this “dynamical many-body topology” (DMBT). We show numerically that both the pure state and ensemble signatures are remarkably robust, and argue that these nonequilibrium signatures should be quite generic in SPT systems, regardless of protecting symmetries or spatial dimension.

We investigate how photoinduced topological phase transitions and the magnetic-field-induced quantum Hall effect simultaneously influence the Casimir force between two parallel sheets of staggered two-dimensional (2D) materials of the graphene family. We show that the interplay between these two effects enables on-demand switching of the force between attractive and repulsive regimes while keeping its quantized characteristics. We also show that doping these 2D materials below their first Landau level allows one to probe the photoinduced topology in the Casimir force without the difficulties imposed by a circularly polarized laser. We demonstrate that the magnetic field has a huge impact on the thermal Casimir effect for dissipationless materials, where the quantized aspect of the energy levels leads to a strong repulsion that could be measured even at room temperature.

We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of unity, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the shift capacitor coupling between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, topolectric circuits, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables us to implement a wide class of topological models through rather simple topolectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, and second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the stacked topolectric circuit construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac, and Weyl semimetals, respectively, displaying surface- and hinge-localized impedance.

Majorana corner modes appearing in two-dimensional second-order topological superconductors have great potential applications for fault-tolerant topological quantum computations. We demonstrate that in the presence of an in-plane magnetic field, two-dimensional ($s+p$)-wave superconductors host Majorana corner modes, whose location can be manipulated by the direction of the magnetic field. In addition, we discuss the effects of edge imperfections on the Majorana corner modes. We describe how different edge shapes and edge disorder affect the number and controllability of the Majorana corner modes, which is of relevance for the implementation of topological quantum computations. We also discuss tunneling spectroscopy in the presence of the Majorana corner modes, where a lead wire is attached to the corner of the noncentrosymmetric superconductor. The zero-bias differential conductance shows a distinct periodicity with respect to the direction of the magnetic field, which demonstrates the excellent controllability of the Majorana corner modes in this setup. Our results lay the theoretical groundwork for observing and tuning Majorana corner modes in experiments on ($s+p$)-wave superconductors.

By means of first-principles calculations and crystal structure searching techniques, we predict that a different NaCaBi family crystallized into the ZrBeSi-type structure (i.e., $P{6}_3/mmc$) are strong topological insulators (STIs). Taking $P{6}_3/mmc$ NaCaBi as an example, the calculated band structure indicates that there is a band inversion between two opposite-parity bands at the $\mathrm{\Gamma }$ point. In contrast to the well-known ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ family, the band inversion in the NaCaBi family has already occurred even without spin-orbit coupling (SOC), giving rise to a nodal ring surrounding $\mathrm{\Gamma }$ in the $k_z=0$ plane (protected by $M_z$ symmetry). With time-reversal symmetry $\left(\mathcal{T}\right)$ and inversion symmetry $\left(\mathcal{I}\right)$, the spinless nodal-line metallic phase protected by ${\left[\mathcal{TI}\right]}^2=1$ is the weak-SOC limit of the spinful topological insulating phase. Upon including SOC, the nodal ring is gapped, driving the system into a STI. Besides inversion symmetry, the nontrivial topology of NaCaBi can also be indicated by $\overline{6}$ symmetry. More surprisingly, the SOC-induced band gap in NaCaBi is about 0.34 eV, which is larger than the energy scale of room temperature. Four other compounds (KBaBi, KSrBi, RbBaBi, and RbSrBi) in the family are stable at ambient pressure, both in thermodynamics and lattice dynamics, even though their gaps are smaller than that of NaCaBi. Thus, they provide good platforms to study topological states both in theory and experiments.

We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-André-Harper on-site potential. Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial. Above the critical pairing, topological edge modes are observed in the central gap. For sufficiently fast decay of the pairing, these modes are identified as Majorana zero modes. However, if the pairing term decays slowly, the modes become massive Dirac modes. Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an intimate relation to Majorana physics. We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system. In addition to the central gap around zero energy, the Aubry-André-Harper potential also leads to other energy gaps at nonzero energy. As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant. However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore. This provides another example of the weakening of the bulk-boundary correspondence occurring in this model. Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.

The search for materials with topological properties is an ongoing effort. In this article we propose a systematic statistical method, supported by machine learning techniques, that is capable of constructing topological models for a generic lattice without prior knowledge of the phase diagram. By sampling tight-binding parameter vectors from a random distribution, we obtain data sets that we label with the corresponding topological index. This labeled data is then analyzed to extract those parameters most relevant for the topological classification and to find their most likely values. We find that the marginal distributions of the parameters already define a topological model. Additional information is hidden in correlations between parameters. Here we present as a proof of concept the prediction of the Haldane model as the prototypical topological insulator for the honeycomb lattice in Altland-Zirnbauer (AZ) class A. The algorithm is straightforwardly applicable to any other AZ class or lattice, and could be generalized to interacting systems.

New ideas for low-mass dark matter direct detection suggest that narrow band gap materials, such as Dirac semiconductors, are sensitive to the absorption of meV dark matter or the scattering of keV dark matter. Here we propose spin-orbit semiconductors—materials whose band gap arises due to spin-orbit coupling—as low-mass dark matter targets owing to their $O$(10 meV) band gaps. We present three material families that are predicted to be spin-orbit semiconductors using density functional theory (DFT), assess their electronic and topological features, and evaluate their use as low-mass dark matter targets. In particular, we find that the tin pnictide compounds are especially suitable having a tunable range of meV-scale band gaps with anisotropic Fermi velocities allowing directional detection. Finally, we address the pitfalls in the DFT methods that must be considered in the ab initio prediction of narrow-gapped materials, including those close to the topological critical point.

We build exactly solvable lattice Hamiltonians for fermionic symmetry-protected topological (SPT) phases in ($3+1$)D classified by group supercohomology. A central benefit of our construction is that it produces an explicit finite-depth quantum circuit (FDQC) that prepares the ground state from an unentangled symmetric state. The FDQC allows us to clearly demonstrate the characteristic properties of supercohomology phases—namely, symmetry fractionalization on fermion parity flux loops—predicted by continuum formulations. By composing the corresponding FDQCs, we also recover the stacking relations of supercohomology phases. Furthermore, we derive topologically ordered gapped boundaries for the supercohomology models by extending the protecting symmetries, analogous to the construction of topologically ordered boundaries for bosonic SPT phases. Our approach relies heavily on dualities that relate certain bosonic 2-group SPT phases with supercohomology SPT phases. We develop physical motivation for the dualities in terms of explicit lattice prescriptions for gauging a 1-form symmetry and for condensing emergent fermions. We also comment on generalizations to supercohomology phases in higher dimensions and to fermionic SPT phases outside of the supercohomology framework.

The celebrated tenfold way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of nonspatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly interacting electrons' noncrystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct ground states as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold way of Altland-Zirnbauer nonspatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold way.

We study the spin transport through a 1D quantum Ising-XY-Ising spin link that emulates a topological superconducting-normal-superconducting structure via Jordan-Wigner (JW) transformation. We calculate, both analytically and numerically, the spectrum of spin Andreev bound states and the resulting ${\mathbb{Z}}_2$ fractional spin Josephson effect (JE) pertaining to the emerging Majorana JW fermions. Deep in the topological regime, we identify an effective time-reversal symmetry that leads to ${\mathbb{Z}}_4$ fractional spin JE in the presence of interactions within the junction. Moreover, we uncover a hidden inversion time-reversal symmetry that protects the ${\mathbb{Z}}_4$ periodicity in chains with an odd number of spins, even in the absence of interactions. We also analyze the entanglement between pairs of spins by evaluating the concurrence in the presence of spin current and highlight the effects of the JW Majorana states. We propose to use a microwave cavity setup for detecting the aforementioned JEs by dispersive readout methods and show that, surprisingly, the ${\mathbb{Z}}_2$ periodicity is immune to any local magnetic perturbations. Our results are relevant for a plethora of spin systems, such as trapped ions, photonic lattices, electron spins in quantum dots, or magnetic impurities on surfaces.

We analyze the robustness of corner modes in topological photonic crystals, taking a $C_6$-symmetric breathing honeycomb photonic crystal as an example. First, we employ topological quantum chemistry and Wilson loop calculations to demonstrate that the topological properties of the bulk crystal stem from an obstructed atomic limit phase. We then characterize the topological corner modes emerging within the gapped edge modes employing a semianalytical model, determining the appropriate real-space topological invariants. We provide a detailed account of the effect of long-range interactions on the topological modes in photonic crystals, and we quantify their robustness to perturbations. We conclude that, while photonic long-range interactions inevitably break chiral symmetry, the system is reducible to a chirally symmetric limit and the corner modes are protected by this together with lattice symmetries.

In this work, we report a ubiquitous presence of topological Floquet time crystal ($\mathrm{TFTC}$) in one-dimensional periodically driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry ($\mathrm{DTS}$) breaking in our $\mathrm{TFTC}$ model require necessarily coexistence of anomalous topological invariants (0 modes and $\pi$ modes), instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between 0 and $\pi$ modes can produce the period doubling ($2T$, two cycles of the drive) that breaks the $\mathrm{DTS}$ spontaneously, leading to the subharmonic response ($\omega /2$, half the drive frequency). The rigid period-$2T$ oscillation is topologically protected against perturbations due to both nontrivially opening of 0 and $\pi$ gaps in the quasienergy spectrum, thus, as a result, can be viewed as a specific “Rabi oscillation” between two Floquet eigenstates with certain quasienergy splitting $\pi /T$. Our modeling of the time-crystalline “ground state” can be easily realized in experimental platforms such as topological photonics and ultracold fields. Also, our work can bring significant interest to explore topological phase transition in Floquet systems and to bridge the gap between Floquet topological insulators and photonics, and period-doubled time crystals.

We consider a mesoscopic superconducting island hosting multiple pairs of Majorana zero-energy modes. The Majorana island consists of multiple $p$-wave wires connected together by a trivial ($s$-wave) superconducting backbone and is characterized by an overall charging energy $E_C$; the wires are coupled to normal-metal leads via tunnel junctions. We calculate the average charge on the island as well as nonlocal conductance matrix as a function of a $p$-wave pairing gap ${\mathrm{\Delta }}_P$, charging energy $E_C$, and dimensionless junction conductances $g_i$. We find that the presence of a topological ground-state degeneracy in the island dramatically enhances charge fluctuations and leads to the suppression of Coulomb blockade effects. In contrast with conventional ($s$-wave) mesoscopic superconducting islands, we find that Coulomb blockade oscillations of conductance are suppressed in Majorana islands regardless of the ratio $E_C/{\mathrm{\Delta }}_P$ or the magnitude of the conductances $g_i$. We also discuss our findings in relation to the so-called topological Kondo effect.

We propose a minimal lattice model for two-dimensional class DIII superconductors with $C_2$-protected higher-order topology. Although this class of superconductors cannot be topologically characterized by symmetry eigenvalues at high-symmetry momenta, we propose a simple Wannier-orbital-based real-space diagnosis to unambiguously capture the corresponding higher-order topology. We further identify and characterize a variety of conventional topological phases in our minimal model, including a weak topological superconductor and a nodal topological superconductor with chiral-symmetry protection. The disorder effect is also systematically studied to demonstrate the robustness of higher-order bulk-boundary correspondence. Our theory lays the groundwork for predicting and diagnosing $C_2$-protected higher-order topology in class DIII superconductors.

Non-Hermitian skin effect and critical skin effect are unique features of non-Hermitian systems. In this work, we explore novel phenomena associated with the ${\mathbb{Z}}_2$ skin effect and critical skin effect in open quantum systems. For the open system with the corresponding damping matrix exhibiting ${\mathbb{Z}}_2$ skin effect, we identify the existence of helical damping, characterized by the evolution of relative particle number with exponentially and algebraically decreasing intervals distinguished by sharp wave fronts with opposite propagation directions. When the time-reversal symmetry is broken by adding perturbations, we find the occurrence of dynamical critical skin effect, which causes the disappearance of the helical damping in the thermodynamic limit although it can survive in small size systems. We also demonstrate the existence of anomalous critical skin effect when we couple two identical systems with ${\mathbb{Z}}_2$ skin effect. With the help of non-Bloch band theory, we unveil that the change of generalized Brillouin zone equation is the origin of critical skin effect.

Based on a topological transition of the symmetry-protected topological phase (SPT), an interaction-induced topological charge pump (iTCP) is proposed with the symmetry-breaking parameter as a synthetic dimension. It implies that the phase boundary of the SPT is a topological obstruction, although iTCP and the gap closing singularity are stable for symmetry-breaking perturbations. As for the iTCP, an interaction is essential since the pumped charge is trivial for a noninteracting system. We have confirmed the bulk-edge correspondence for this iTCP using density matrix renormalization group for the Rice-Mele model with nearest-neighbor interactions. As for a realization in optical lattices, an interaction sweeping pump protocol is proposed as well.

We demonstrate the coexistence of pseudospin- and valley-Hall-like edge states in a photonic crystal with $C_{3v}$ symmetry, which is composed of three interlacing triangular sublattices with the same lattice constants. By tuning the geometry of the sublattices, three complete photonic band gaps with nontrivial topology can be created, one of which is due to the band inversion associated with the pseudospin degree of freedom at the $\mathrm{\Gamma }$ point and the other two due to the gapping out of Dirac cones associated with the valley degree of freedom at the $K,K^{\prime }$ points. The system can support triband pseudospin- and valley-momentum locking edge states at properly designed domain-wall interfaces. Furthermore, to demonstrate the novel interplay of the two kinds of edge states in a single configuration, we design a four-channel system, where the unidirectional routing of electromagnetic waves against sharp bends between two routes can be selectively controlled by the pseudospin and valley degrees of freedom. Our work combines the pseudospin and valley degrees of freedom in a single configuration and may provide more flexibility in manipulating electromagnetic waves with promising potential for multiband and multifunctional applications.

We develop a theory of edge states based on the Hermiticity of Hamiltonian operators for tight-binding models defined on lattices with boundaries. We describe Hamiltonians using shift operators which serve as differential operators in continuum theories. It turns out that such Hamiltonian operators are not necessarily Hermitian on lattices with boundaries, which is due to the boundary terms associated with the summation by parts. The Hermiticity of Hamiltonian operators leads to natural boundary conditions, and for models with nearest-neighbor (NN) hoppings only, there are reference states that satisfy the Hermiticity and boundary conditions simultaneously. Based on such reference states, we develop a Bloch-type theory for edge states of NN models on a half-plane. This enables us to extract Hamiltonians describing edge states at one end, which are separated from the bulk contributions. It follows that we can describe edge states at the left and right ends separately by distinct Hamiltonians for systems of cylindrical geometry. We show various examples of such edge-state Hamiltonians, including Hofstadter model, graphene model, and higher-order topological insulators.